Chris Samba had given Sam Allardyce's Rovers the half-time lead in controversial circumstances on Saturday, while Fulham had barely had a sniff of goal.
The second half was a different story, however, as Mark Hughes' men had much the better of the match and secured a scoreline that was on balance a fair result.
Blackburn were unchanged with 18-year-old Phil Jones, who played with distinction as the holding midfield player against Manchester City, impressing again in a different role to his usual position in central defence.
Former Rovers player and manager Hughes chose an old Blackburn team-mate Damien Duff to take the place of Bobby Zamora, who suffered a broken leg last week.
Blackburn were utterly dominant for much of the opening half, starting when Nikola Kalinic won an early free-kick in a dangerous spot right on the corner of the area - another inch and it would have been a penalty - but Fulham were able to clear.
Fulham goalkeeper Mark Schwarzer came close to having a nightmare afternoon with a series of fumbles. The first nearly led to the opening goal but Brede Hangeland managed to head clear off the line after his fellow Norwegian - and good friend - Morten Gamst Pedersen delicately chipped in the loose ball.
Jones hooked over from the ensuing long throw-in and, when Fulham did finally conjure up a meaningful opportunity, Dempsey drifted the direct free-kick well wide.
Mame Biram Diouf looked to be in on goal but his control deserted him at the critical point, and the Senegalese striker then claimed a penalty when he was challenged by Hangeland, though it appeared the big Norwegian just nicked the ball away.
Hangeland was then booked for going through the same player and, from the free-kick, the goal that Blackburn had been threatening finally arrived.
Paul Robinson hoisted the ball high and with Schwarzer again looking uncertain - and some shoving by El-Hadji Diouf not helping Fulham's cause - Samba won the heading lottery to loop the ball into the empty net.
Hughes reacted with anger on the touchline, provoking good-natured jeers from the home crowd, and his mood would not have been improved when Robinson displayed superb agility to turn aside Carlos Salcido's impressing curling strike.
Fulham looked much more positive in the second half and had a chance to get back on terms within seconds of the re-start but Simon Davies' direct free-kick was woeful.
Dempsey made no mistake in the 56th minute, however, when Salcido floated across an inviting cross from left, beating Ryan Nelsen to the ball to plant a powerful header past Robinson for the equaliser.
Schwarzer dropped the ball again under pressure from Samba as Blackburn tried to respond and this time referee Anthony Taylor awarded a free-kick when replays showed clearly there had been no foul.
Benjani Mwaruwari came on for Kalinic to make his Blackburn debut and his header down gave Brett Emerton space for a snap-shot but he hit it straight at Schwarzer.
Schwarzer's uncertainty continued to be a problem for the visitors and he was left grateful to his defence for keeping out El-Hadji Diouf after another fumble.
Fulham were lucky to survive again when Salgado whipped a vicious ball across the face of goal, then Danny Murphy tricked the Spanish defender into conceding a free-kick on the edge of his box.
Murphy tried to curl it into the top corner but Robinson never looked worried as it sailed over.
The match might have been there for the taking for either side after that, but fear of defeat won the day as the game drifted into a stalemate.